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Let $\left(Z, *\right)$ be an algebraic structure where $Z$ is the set of integers and the operation $*$ is defined by $n*m = \max(n,m)$. Which of the following statements is true for $\left(Z, *\right)$?

- $\left(Z, *\right)$ is a monoid
- $\left(Z, *\right)$ is an Abelian group
- $\left(Z, *\right)$ is a group
- None of the above

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32 votes

Best answer

Lets follow our checklist one by one to see what property this algebraic structure follows.

Closure -yes ($m*n=\max(m,n)$) Output is either $m$ or $n$ whichever is maximum and since $m,n$ belongs to $Z$, the result of the binary operation also belongs to $Z$. So closure property is satisfied.

Associative-Yes the output is max among the elements and it is associative

Now for identity, we don't have a single unique element for all the elements which is less than all the elements. ie, $m*e=m \implies \max(m,e) = m$. We can't find a single unique $e$ which is less than all possible integer $m$, such that comparison between the two would always give $m$ itself.

If the set was for Natural numbers we could have had 1 as identity element as it is less than any other natural number and we would have got a monoid.

- Semi-group - Closed and associative
- Monoid - Closed, associative and has an identity
- Group - Monoid with inverse
- Abelian group- Group with commutative property.

Hence, this is just a semigroup- D option.

Ans D)

4 votes

Given (Z,*) is an algebraic structure. Hence it is closed.

Now, to become SEMI-GROUP: It should be associative

(x*y)*z = max(max(x,y),z) = max(x,y,z)

x*(y*z) = max(x,max(y,z)) = max(x,y,z)

eg. (1*2)*3 = max((max(1,2),3) = max(2,3) = 3

1*(2*3) = max(1,max(2,3)) = max(1,3) = 3

Hence (x*y)*z = x*(y*z), so ASSOCIATIVE i.e SEMI-GROUP

Now, to become MONOID: It should have an identity element

Let there be such an element and let the identity element be e.

So, as per definition: (a*e) = max(a,e) = a

Nice. But the question is: What if we take e=(a+1) ∀ a ∈ Z

Then, (a*(a+1)) = max(a,(a+1)) = (a+1) and **NOT a**

So, there can be no such identity(e) element possible.

Hence, its not a MONOID.

So correct choice is: (D) None

4 votes

Closure Property => This is satisifed .Maximum of two integer is one of the integers. So we satisfy closure property.

Associativity => This property also satisfied.

2,3,4 =Max(2(max(3,4)) = 4 = Max(Max(2,3),4) .

Identity => We do not have identity.

Suppose some i is identity.

Then Max(i,anything) should be that anything !

But if I take i-1. (As we have Z , If i is integer so is i-1)

Max(i,i-1) = i. So i is not identity.

So This is not monoid. & SO not group or abelian group . Ans is D

Associativity => This property also satisfied.

2,3,4 =Max(2(max(3,4)) = 4 = Max(Max(2,3),4) .

Identity => We do not have identity.

Suppose some i is identity.

Then Max(i,anything) should be that anything !

But if I take i-1. (As we have Z , If i is integer so is i-1)

Max(i,i-1) = i. So i is not identity.

So This is not monoid. & SO not group or abelian group . Ans is D

3 votes

Hi , in this question does identity element always exist ? because , if I take a set A = {-2,-1,0,1,2}

( 0 can be included as it is integer set ).

so , -2 * e = -2 ( where e is the identity element ) , but if here we can not have any such unique identity element .

So , this is not going to be Monoid .

Please correct me , if I am wrong.

( 0 can be included as it is integer set ).

so , -2 * e = -2 ( where e is the identity element ) , but if here we can not have any such unique identity element .

So , this is not going to be Monoid .

Please correct me , if I am wrong.